$\arctan$ with two arguments

When inverting the polar coordinates, one needs thearc tan function (http://planetmath.org/CyclometricFunctions) $\arctan$with two arguments.If $(x,y)\in {ℝ}^2\setminus \{0\}$, then$\arctan (x,y)$is defined as the angle $(x,y)$ makes withthe positive $x$-axis.

One usually sees expressions like $\arctan (y/x)$,which is equal to $\arctan (x,y)$ when $(x,y)$ is in the first quadrant.However, $\arctan (y/x)$ does not give the correct angle when $(x,y)$is in the third quadrant (since $y/x=(-y)/(-x)$).Also, the quotient $y/x$ involves a division by zero when $x=0$,which is damaging both numerically and mathematically.

In most mathematical software and programming languages the two-argument $\arctan$is directly implemented.

In Python language the functions atan(x) and atan2(x,y) are the respective one and two argument versions of $\arctan$. The point of having the two argument version is to determine the correct quadrant of the point. For instance, $1/1=1=-1/-1$, so atan(x) cannot distinguish between $(1,1)$ and $(-1,-1)$, but atan2(x,y) can, as the following Python code illustrates:

\\PMlinkescapetext{>>> from math import *>>> print atan(1)0.785398163397>>> print atan2(1,1)0.785398163397>>> print atan2(-1,-1)-2.3619449019}

because $(1,1)$ has argument $\pi /4=0.7853\mathrm{\dots }$ but $(-1,-1)$ has argument $-3\pi /4=-2.3619\mathrm{\dots }$.

Analytic properties

In mathematical works, $\arctan (x,y)$is simply denoted by $\theta (x,y)$. The symbol $\theta$ obviously refers to the angle, but it is reallythe function $h_2$, where

$$g(r,\theta )=(r\cos \theta ,r\sin \theta ),h(x,y)=g^{-1}(x,y)=(r,\theta ).$$

The function $g:{ℝ}^2\to {ℝ}^2$ is the polar-to-Cartesian coordinate transformation.By the inverse function theorem, the function $h$ (the Cartesian-to-polar coordinate transformation) exists and is smooth wherever it is defined.Note that $h$ cannot be defined continuously everywhere, because of the multi-valued nature of $\theta$ — $(r,\theta )$ and $(r,\theta +2\pi n)$ always map to the same point under $g$.(Similarly, $\theta$ cannot defined when $r=\sqrt{x^2+y^2}=0$.)This means, if one chases a loop (say a circle) around the origin, $\theta$ would movefrom $0$ to $2\pi$, even though the image point $g(r,\theta )$ winds back to the starting point.

Technically, a “largest” possible domain of $h$ (and $\theta$) can only be taken to be some simply connected open subset of ${ℝ}^2\setminus \{0\}$. (Note: ${ℝ}^2\setminus \{0\}$ itself is not simply connected.)For example, such a domain might be ${ℝ}^2\setminus \{(x,y):x\le 0\}$, i.e. delete the negative real axis from ${ℝ}^2$.

The exterior derivative of $\theta$is

$$d\theta =\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy,$$

(found by implicit differentiation),and hence

$$\frac{\partial \theta }{\partial x}=\frac{-y}{x^2+y^2},\frac{\partial \theta }{\partial y}=\frac{x}{x^2+y^2}$$

(which can also be found by differentiating $\arctan (y/x)$ directlyand piecing the results for each quadrant).

Of course, the formulas above are only valid wherever $\theta$ is defined,but the analytical expressions do not change no matter which domain of definition is taken for $\theta$.This allows for the following neat formulato find the total variation of angle of a smooth curve $\gamma :[a,b]\to {ℝ}^2\setminus \{0\}$:

$${\int }_{\gamma }𝑑\theta ={\int }_a^b{\gamma }^{*}𝑑\theta ={\int }_a^b\left(\frac{-y\dot{x}}{x^2+y^2}+\frac{x\dot{y}}{x^2+y^2}\right)𝑑t.$$

(This is related to the formula for the winding numberand the argument principle in complex analysis.)

For example, if $\gamma (t)=(r\cos t,r\sin t)$, for $t\in [0,2\pi n]$,is the circle that winds around the origin $n$ times, then ${\int }_{\gamma }𝑑\theta =2\pi n$.